(1) Suppose that \(N_1\subset N_2\) . Let \(\tilde{\ell }_1\)  and \(\tilde{\ell }_2\)  be word length functions on \(N_1\)  and \(N_2\) , respectively. Then \(\tilde{\ell }_2\lesssim \tilde{\ell }_1\)  on \(N_1\) .
We claim that then \(\log (1+ \tilde{\ell _2}\,)\lesssim \log (1+ \tilde{\ell _1}\,)\)  on \(N_1\) . Indeed, take \(C,D>0\)  such that
\(\tilde{\ell }_2(g)\le C\,\tilde{\ell }_1(g)+D\)  for all \(g\in N_1\) . Fix \(g\)  and note that
\(\log (1+ \tilde{\ell }_2(g))\le \log (1+ C\,\tilde{\ell }_1(g)+D)\le \log (1+D)+\log (1+ k\,\tilde{\ell }_1(g)),\) 
where \(k\)  is an integer satisfying \(k\ge (1+D)^{-1}C\) . Since \(1+ kt\le (1+t)^k\)  for every \(t\ge 0\) , we conclude that \(\log (1+ \tilde{\ell _2}\,)\lesssim \log (1+ \tilde{\ell _1}\,)\)  on \(N_1\) .
Further, \(\ell _2 \lesssim \log (1+ \tilde{\ell }_2\,)\)  on \(N_2\)  since \(\ell _2\)  is exponentially distorted on \(N_2\) . Hence \(\ell _2 \lesssim \log (1+ \tilde{\ell }_1\,)\)  on \(N_1\) , i.e., \(\ell _2\)  is exponentially distorted on \(N_1\) . So being a maximal length function exponentially distorted on \(N_1\) , the function \(\ell _1\)  dominates \(\ell _2\)  on the whole \(G\) .
(2) Suppose that \(N_1\)  is not contained in \(N_2\) , i.e., \(N_1\ne N_1\cap N_2\) . Denote the Lie algebra of \(N_j\) , \(j=1,2\) , by \(\mathfrak {n}_j\) . Since \(N_j\)  is nilpotent, closed and simply connected (Lemma REF ), we have that \(\mathfrak {n}_1\ne \mathfrak {n}_1\cap \mathfrak {n}_2\) . Then there is \(\eta \in \mathfrak {n}_1\setminus (\mathfrak {n}_1\cap \mathfrak {n}_2)\) .
Let \(\mathfrak {b}\)  and \(\mathfrak {h}\)  be as above. We claim that \(\eta \)  can be taken in a subspace \(\mathfrak {v}_2\)  of \(\mathfrak {h}\)  complementary to \(\mathfrak {n}_2\cap \mathfrak {h}\) . Indeed,
since \(\mathfrak {e}\subset \mathfrak {n}_1\cap \mathfrak {n}_2\) , we have \((\mathfrak {n}_1\cap \mathfrak {n}_2)+\mathfrak {h}=\mathfrak {b}\)  (cf. the proof of Theorem REF ).
Evidently \(\eta \in \mathfrak {b}\)  and so there are \(\eta ^{\prime }\in \mathfrak {n}_1\cap \mathfrak {n}_2\)  and \(\eta ^{\prime \prime }\in \mathfrak {h}\)  such that \(\eta =\eta ^{\prime }+\eta ^{\prime \prime }\) . Then \(\eta ^{\prime }\in \mathfrak {n}_1\)  and also \(\eta ^{\prime \prime }\notin \mathfrak {n}_2\)  because otherwise \(\eta \in \mathfrak {n}_2\) . Since \(\eta ^{\prime \prime }\)  is in \(\mathfrak {h}\)  but not in \(\mathfrak {n}_2\cap \mathfrak {h}\) , we can take \(\mathfrak {v}_2\)  containing \(\eta ^{\prime \prime }\) . The claim is proved.
Note that \(\mathfrak {b}/\mathfrak {n}_2\)  is nilpotent. It is not hard to see (see the proof of Proposition 4.1 in [1]}) that there is an \({F}\) -basis (i.e., compatible with the lower central series) \((e_k)\)  in \(\mathfrak {b}/\mathfrak {n}_2\)  containing the element \(\eta +\mathfrak {n}_2\) .
By Theorem REF , \(\ell _j\) , \(j=1,2\) , is equivalent to the function defined in (REF ) with \(N^{\prime }=N_j\) . To use this equivalence we need the following asymptotic for a word length functions on \(B/N_j\) : it is equivalent to the function \(\max _k|t_k|^{1/w_k}\) , where \((t_k)\)  are the canonical coordinated of the first kind associated with the \({F}\) -basis and \(w_k\in \mathbb {N}\) , [1]}. Note that the quotient map \(\mathfrak {b}\rightarrow \mathfrak {b}/\mathfrak {n}_2\)  induces a bijection \(\mathfrak {b}\rightarrow \mathfrak {v}_2\) . Putting \(j=2\)  and taking \(k\)  such that \(e_k=\eta +\mathfrak {n}_2\)  we have from (REF ) that
\(\ell _2(\exp (t \eta )) \simeq |t|^{1/w_k},\) 
where \(t\in \mathbb {C}\) . On the other hand, \(\eta \in \mathfrak {n}_1\)  and so (REF ) written for \(j=1\)  gives
\(\ell _1(\exp (t \eta )) \simeq \log (1+|t\eta |),\) 
where \(|\cdot |\)  a norm on \(\mathfrak {n}\) . Thus \(\ell _2\)  is not dominated by \(\ell _1\) .
